Download - Solar Acoustic Holograms

Tsing Hua University, Taiwan

Solar Acoustic Holograms

January 2008, Tucson

Dean-Yi Chou

Motivation

Is it feasible to apply the principle of optical holography to a system of solar acoustic waves and active regions?

Contents

Principle of optical holography.

Concept of acoustic holography of active regions.

Construct 3-D wave fields of the magnetic region from the acoustic hologram.

Set up a simplified model to compute acoustic holograms of magnetic regions.

1. analogies and differences between two

2. difficulties

Challenges and prospects.

Hologram (interference pattern)

(time average)

Construction of Waves

hologram

diffraction field

(Gabor’s in-line holgram)

Acoustic waves on the Sun

solar surface

interference pattern

Solar Acoustic Waves + Active Region

(acoustic power map)

perturbed region

2

0 sI

Optical Holography Solar Acoustic Holography

reference wave

object

hologram

p-mode wave

magnetic region

acoutsic power map

Analogies

(coming from below)

(near the surface)

(on the surface)

Questions:

1. Can we detect the inference pattern (hologram) due to a magnetic region on the surface?

2. Can we use the observed hologram to construct the 3-D image of the magnetic region?

Optical Holography Solar Acoustic Holography

1. monochromatic

5. far field approximation

4. single reference wave

finite band width

wavelength

~ dimension of object

~ distance to hologram

* multiple incident waves

Differences

2. no boundary trapped in cavities

3. straight ray path curved ray path

If the width of power spectrum of a wave field is , the cohernt time of waves is

coherent time of waves

0

2

1

2

1

0

: central frequency

: period of central frequency

example

0 3.3 mHz

0.2 mHz (FWHM = 0.47 mHz)

2.6

solar surface

trapped in cavitiescurved ray pathmultiple incident waves

2. Waves are approximately vertical near the surface

1. Refracted waves from the lower turning point are ignored.

s

a

λ ～ a ～ s

Multiple Incident Waves

i

i0If incident waves are , total waves are )( 0 sii

i

Intensity of hologram

jijsisjisjsiji

i i iisisii

sjj

jsii

i

*0

*0

**00

*0

22

0

**00

Re2

)()(

cross terms

If different waves are uncorrelated, the contribution from cross terms is small.

Total interference is the sum of interference of individual wave.

interference term

Summation of interferences of different waves reduces the visibility of fringes.

1. Set up a simplified model for scattering of acoustic waves by a magnetic region.

2. Solve for the scattered waves.

3. Compute the interference pattern (hologram) between incident wave and scattered wave.

4. Study the influence of various parameters on the hologram.

5. Compute the constructed wave field by illuminating the hologram with a reference wave.

Model Study

Assume unperturbed medium is uniform, and the wave equation is

Assume the interaction between waves and magnetic regions is described by sound-speed perturbations:

0),(1

),(2

2

22

t

tx

ctx

)()( 10 xccxc

),(

),(12

),(1),(

2

2

20

12

2

22

txS

t

tx

cc

c

t

tx

ctx

time independent

Wave equation becomes

Source of scattering

Wave Equation

Solution of Scattered Wave

'|'

)','(1

'

/)'(

2

1

'')','()',';,(),(

3/''2

02

20

01

3

xdt

tx

cxx

cxc

dtxdtxStxtxGtx

cxxtt

s

),(),(),( 0 txtxtx s

scattered wave with Green’s function and Born approximation

wave equation

total solution

t

tx

cc

ctxS

t

tx

ctx

2

2

20

12

2

22 ),(1

2),(),(1

),(

expressed in terms of Fourier components

dextx ti ),(2

1),( 00

dxdex

cxx

cxctx cxxti

s '),'('

/)'(

2

1),( 3)/'(

020

201

2/3

Hologram

Intensity of the hologram is the time average of 2

*0

22

0**

00

2Re2 ssss

interference

0/'2/

2/

*002

0

2013

*0

2/

2/

*0

),(),'('

/)'('

2

1

),(),(1

cxxiT

T

T

T

ss

exxc

dxx

cxcxd

T

dttxtxT

Interference term

Need a model for spatial dependence of ),(0 x

A Simplified Model for ),(0 x

assumptions:

1. Consider only one upward wave mode and its reflected wave at the surface.

2. Assume the free-end boundary at the surface.

)(0

)(00 )()(),( zkxkizkxki zz eRex

1R

xkiz ezkx

)cos()(2),( 00

interference term

]/')'([2

020

2013*

00)'cos()cos()(

'

/)'('

2),'(),( cxxxxki

zzs ezkzkc

dxx

cxcxd

Ttxtx

normalized interference term (related to fringe visibility)

2

0

]/')'([2

020

2013

2

0

*0

)(

)'cos()cos()('

/)'('

2Re

)(

),(),(Re2

0

d

ezkzkc

dxxcxc

xdT

t

txtx

cxxxxkizz

s

3. Simple dispersion relation:

)( 22

20

2zkkc

Normalized Interference Term (fringe visibility)

2

0

]/')'([2

020

2013

2

0

*0

)(

)'cos()cos()('

/)'('

2Re

)(

),(),(Re2

0

d

ezkzkc

dxxcxc

xdT

t

txtx

cxxxxkizz

s

Effects of parameters on holograms

1. coherent time of incident waves

3. size of the perturbed region

4. depth of the perturbed region

2. wavelength

5. angle of incidence

0

2

1

2

1

0

Effects of Coherent Time of Incident Waves

Setup of incident wave

3. Modes with a Gaussian power spectrum centered at 3.3 mHz, with different widths.

1. Waves propagate vertically: 0k

2. Dispersion relation: 2

22 kc

4. coherent time

Perturbed region

1. Uniform cylinder with 03.0/ 01 cc

2. diameter = 9.6 Mm, vertical extent = 4.8 Mm, depth = 12 Mm

3.3 mHz, 14.7 Mm (l=300), 0c 48.5 km/s

Effects of Coherent Time

0.2 mHz (FWHM = 0.47 mHz)

line width

Effects of Wavelength

0 3.3 mHz,

0.2 mHz

uniform cylinder with 03.0/ 01 cc

diameter = 9.6 Mm, vertical extent = 4.8 Mm, depth = 12 Mm

wavelength

Effects of Angle of Incidence

At 5Mm depth, the angle of incidence is about for at 3.3 mHz.010 100l020 for at 3.3 mHz.200l

Waves with different phase velocities have different angles of incience.

For example:

Effects of Angle of Incidence (cont.)

0 3.3 mHz, 0.2 mHz,

uniform cylinder with 03.0/ 01 cc

diamter = 9.6 Mm, vertical extent = 4.8 Mm, depth = 12 Mm

14.7 Mm (l=300)

incident angle

Construction of Wave Fields from Holograms

Illuminate the hologram by a vertically-propagating monochromatic wave.

hologram on the surface

Advantages of digital holograms

DC signal

2. Disentangling wave fields of virtual and real images.

1. DC signals are removed to enhance the interference pattern.

hologram on the surface

')'('

)'('

'1

'2)(

'

daxxx

xxn

xxk

i

xx

e

i

kx

xxik

Diffraction waves are computed by the Kirchhoff intergral

replaced by

Constructed wave field

205 Mm

30 Mm

7.14

Incident angle = 0

Mm

depth = 30 Mm


Incident angle = 0 deg.

Depth = 30 Mm


Depth = 12 Mm



Depth = 30 Mm


Depth = 30 Mm

Effects of Multiple Incident Waves

1. Weaken holograms

2. Distort and weaken constructed wave fields

The maximum occurs at . 2/ zyx

dc

c 0

1max

1. Signals of holograms are weak.

Challenges in detecting interference fringes

2. Interference fringes are contaminated by suppression of acoustic power in magnetic region.

Fluctuation of 1000 MDI Dopplergrams is about 10%.

1% for the 2nd and 3rd fringes if

Remove suppression by an empirical relation of power vs. field strength.

Search for interference fringes outside magnetic regions.

3. Find an optimal filter to detect interference fringes.

03.0/ 01 cc

power map before correction power map after correction

magnetic field Power vs. B field

1024 MDI FD images

phase-velocity-filtered power map

magnetic field power map

1024 MDI FD images

phase-velocity-filtered power map

(3.3mHz/300) (3.3mHz/400)

power map before correction power map after correction

magnetic field Power vs. B field

512 MDI HR images

Challenges in Constructed 3D Wave Fields

2. Is there a better way to construct 3D wave fields?

1. How to disentangle wave fields of virtual and real images and obtain the 3D structure of the magnetic region?

Improvement in computing interference fringes

1. A better model to compute scattered waves.

2. Study of simulation data

interaction between waves and B fields

more realistic dispersion relation

Prospects

Better Data

Hinode & HMI